Question

Evaluate $$13$$ to the power of $$-2$$ ?

Answer 1

Hint: In this problem we need to evaluate the value of $$13$$ raised to the power of $$-2$$ . $$13$$ to the negative $$-2$$ power is conventionally written as $${13}^{-2}$$ . Now, we know that $$a^{-n}={\displaystyle \frac{1}{a^n}}$$ .Hence using this property we will first write $${13}^{-2}$$ with positive power. Now we also know that for a positive integer $$n$$ we have $$a^n=a\times a\times a\times ...ntimes$$ . So, we will use this property to evaluate the obtained expression and hence find the value of $${13}^{-2}$$.

Complete step by step answer:
Consider the given number i.e., $${13}^{-2}$$. Now, since we have $$-2$$ power, which is negative power, we will first make this power positive. As, we know that
$$a^{-n}={\displaystyle \frac{1}{a^n}}$$
So, here $$a=13$$ and $$n=2$$
Therefore, we get
$${13}^{-2}={\displaystyle \frac{1}{{13}^2}}$$
Hence, we make this power positive by taking this to the denominator.

Now, we have $${13}^2$$ in the denominator. Now, we also know that for a positive integer $$n$$ we have
$$a^n=a\times a\times a\times ...ntimes$$
So here, $$a=13$$ and $$n=2$$
Therefore, we get as
$${13}^2=13\times 13$$
Now, we know that $$13\times 13=169$$
Hence, we have
$$\therefore {13}^{-2}={\displaystyle \frac{1}{{13}^2}}={\displaystyle \frac{1}{169}}$$

Hence, $$13$$ to the power of $$-2$$ equals $${\displaystyle \frac{1}{169}}$$.

Note: Here the number $$13$$ is called the base, and the number $$-2$$ is called the exponent or index. This question can also be understood in a simple way like: As the exponent is a negative integer, so exponentiation means the reciprocal of a repeated multiplication. The absolute value of the exponent of the number $$-n$$ i.e., $$n$$ denotes how many times we have to multiply the base $$a$$ ,and the power’s minus sign stands for reciprocal.Thus, we have $$13$$ to the power of $$-2$$ which means the absolute value of the exponent of the number $$-2$$ i.e., $$2$$ denotes how many times we have to multiply the base $$13$$ ,and the power’s minus sign stands for reciprocal.Therefore, we get
$${13}^{-2}={\displaystyle \frac{1}{169}}$$